# GMAT数学之光—余数的底层逻辑

r(a) (mod m)

(1) r(a) ± r(b) = r(a ± b)

(2) r(a) * r(b) = r(a * b)

r(8) = r(5+3) = r(5)+r(3) = 2 + 0 = 2 (mod 3)

r(9) = r(5)+r(4) = 2+1 = 3 (mod 3)

2134是否能被3整除？

2000+100+30+4

r(2000+100+30+4) = r(2000)+r(100)+r(30)+r(4)=r(2*1000)+r(1*100)+r(3*10)+r(4*1)

r(2*1000)+r(1*100)+r(3*10)+r(4*1) = r(2)*r(1000)+r(1)*r(100)+r(3)*r(10)+r(4)*r(1)

r(2)*r(1000)+r(1)*r(100)+r(3)*r(10)+r(4)*r(1) = r(2)+r(1)+r(3)+r(4)

r(2)+r(1)+r(3)+r(4) = r(2+1+3+4)

If n is a positive integer and r is the remainder when 4 + 7n is divided by 3, what is the value of r?

(1) n + 1 is divisible by 3.

(2) n > 20

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

r(4+7n) = r(4)+r(7n)= r(4)+r(7)*r(n) = 1+1*r(n) = 1+r(n)

r(n)+ 1 = 0 相当于 r(n)+ 1 = 3，即，r(n) = 2。充分。

If x is an integer greater than 0, what is the remainder when x is divided by 4?

(1) The remainder is 3 when x+1 is divided by 4.

(2) The remainder is 0 when 2x is divided by 4.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

r(x+1) = r(x)+r(1) = r(x)+1 = 3

r(x) = 2；充分。

What is the remainder when the positive integer n is divided by 3?

(1)   The remainder when n is divided by 2 is 1.

(2)   The remainder when n + 1 is divided by 3 is 2.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

r(n+1)=r(n)+r(1)=2

r(n) = 1。充分。

If x and y are positive integers, what is the remainder when 10^x + y is divided by 3?

(1) x = 5

(2) y = 2

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

r(10^x + y) = r(10^x)+r(y)

r(10)=r(1)

r(10)* r(10)=r(1)* r(1)

r(10*10) = r(1*1)

r(100)=r(1)

r(10^x)+r(y) = 1+r(y)

If t is a positive integer and r is the remainder when t^2+5t+6 is divided by 7, what is the value of r?

(1) When t is divided by 7, the remainder is 6.

(2) When t^2 is divided by 7, the remainder is 1.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

r(t^2)=r(t)*r(t) =r(1)=1

What is the remainder when the positive integer n is divided by the positive integer k, where k > 1?

(1) n = (k + 1)^3

(2) k = 5

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

r((k + 1)^3) = r(k+1)* r(k+1)* r(k+1)

If x, y, and z are positive integers, what is the remainder when 100x + 10y + z is divided by 7?

(1) y = 6

(2) z = 3

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

r(100x + 10y + z) = r(100x)+r(10y)+r(z) = r(100)*r(x)+r(10)*r(y)+r(z) = 2*r(x)+3*r(y)+r(z)

0 0 0 0

0 0 0 0

0 0

y = x*q + r （0=

y是被除数，x是除数，q是商，r是余数。

10 = 4*2+2

If r is the remainder when the positive integer n is divided by 7, what is the value of r?

(1) When n is divided by 21, the remainder is an odd number.

(2) When n is divided by 28, the remainder is 3.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

n = 28k+3

r(n) = r(28k)+r(3) = r(28)*r(k) + r(3)

r(n) = 0*r(k) + r(3) = 0+3 = 3

When the integer n is divided by 17, the quotient is x and the remainder is 5. When n is divided by 23, the quotient is y and the remainder is 14. Which of the following is true?

(A) 23x+17y=19

(B) 17x-23y=9

(C) 17x+23y=19

(D) 14x+5y=6

(E) 5x-14y=-6

17x+5 = 23y+14

17x-23y = 9

1. 2的倍数特征：个位数字是偶数。

2. 5的倍数特征：个位是0或者5。

3. 3或9的倍数特征：各个数位之和能被3或9整除。

4. 4的倍数特征：末两位数能被4整除。

5. 8的倍数特征：末三位数能被8整除。

6. 25的倍数的特征：末两位数能被25整除。

7. 125的倍数特征：末三位数能被125整除。

If a six digits 6ab11c is divisible by 8, c =?

(A) 1

(B) 2

(C) 3

(D) 4

(E) 5

8的倍数的特征为“末三位数能被8整除”，112 = 8*14。因此，c = 2，答案为B。

What is the remainder when the two-digit, positive integer x is divided by 3?

(1) The sum of the digits of x is 5.

(2) The remainder when x is divided by 9 is 5.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

r(9k+5) = r(9k)+r(5) = r(9)*r(k)+r(5) = 0*r(k)+r(5) = 0+2 = 2。充分。

7^1除以10余7；7^2除以10余9；7^3除以10余3；7^4除以10余1。

r(7^4) = r(1) (mod 10)

r(7^4)=r(1)

r(7^4) *r(7^4)=r(1) *r(1)

r(7^4)…r(7^4) = r(1) …r(1)

r(7^4…7^4) = r(1…1)

r(7^548) = r(1) = 1

r(7^548) *r(7)= r(1)*r(7)

r(7^549) = r(7) = 7

What is the remainder when 3^24 is divided by 5?

(A) 0

(B) 1

(C) 2

(D) 3

(E) 4

3^1除以5余3；3^2除以5余4；3^3除以5余2；3^4除以5余1。

r(3^4) = r(1)

r(3^24) =r(1)

If n is a positive integer, what is the remainder when 3^(8n+3) + 2 is divided by 5?

(A) 0

(B) 1

(C) 2

(D) 3

(E) 4

3^1除以5余3；3^2除以5余4；3^3除以5余2；3^4除以5余1。

r(3^4) = r(1)

r(3^8n) =r(1)

r(3^(8n+3)) =r(3^8n) *r(3^3) = 1* r(3^3) = 1*r(27) = 2

r(3^(8n+3)+2) = r(3^(8n+3))+r(2) = 2+2 = 4

2除5余2；9除5余4；16除5余1。

When positive integer n is divided by 5, the remainder is 1. When n is divided by 7, the remainder is 3. What is the smallest positive integer k such that k + n is a multiple of 35?

(A) 3

(B) 4

(C) 12

(D) 32

(E) 35

n除以5余1且n除以7余3，我们可以优先确定n的可能性

3除5余3；10除5余0；17除5余2；24除5余4；31除5余1。

When positive integer x is divided by 5, the remainder is 3; and when x is divided by 7, the remainder is 4.  When positive integer y is divided by 5, the remainder is 3; and when y is divided by 7, the remainder is 4.  If x > y, which of the following must be a factor of x - y?

(A) 12

(B) 15

(C) 20

(D) 28

(E) 35

x和y均可以写为18+35k。由于x大于y，所以x-y必为：

18+35m -18 -35n = 35(m-n) （m>n）

When positive integer n is divided by 3, the remainder is 2; and when positive integer t is divided by 5, the remainder is 3. What is the remainder when the product n*t is divided by 15?

(1) n - 2 is divisible by 5.

(2) t is divisible by 3.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

r(n*t) = r[(2+15k)*(3+15m)] = r(6)+r(30m)+r(45k)+r(225k*m) = 6+0+0+0 = 6

1. 普通年能被4整除且不能被100整除的为闰年。（如2004年就是闰年,1900年不是闰年）

2. 世纪年能被400整除的是闰年。(如2000年是闰年，1900年不是闰年)

365/7余1，因此，每过一个普通年，同一日期的星期需向后推一天。

366/7余2，因此，每过一个闰年，同一日期的星期需向后推两天。

June 25, 1982, fell on a Friday.  On which day of the week did June 25, 1987, fall?  (Note: 1984 was a leap year.)

(A) Sunday

(B) Monday

(C) Tuesday

(D) Wednesday

(E) Thursday

1983，1985，1986，1987是普通年，同一日期的星期向后推一天；1984是闰年，同一日期的星期向后推两天，因此，如果1982年6月25日是星期五，则1987年6月25日是星期四（向后加6天）。答案为E。